A154293 -id:A154293 - cp's OEIS frontend

A182155 Integers of the form: 0/3 + 1/3 + 2/3 + 3/3 + 5/3 + 7/3 + 11/3 + 13/3 + 17/3 + ....

0, 1, 2, 6, 14, 26, 66, 94, 147, 264, 663, 759, 916, 1089, 1213, 1343, 1554, 1706, 2113, 2473, 2661, 2861, 3069, 3285, 3513, 3747, 3989, 4497, 4763, 5039, 5323, 5911, 6217, 6527, 6849, 7179, 7690, 8227, 8790, 9566, 9966, 10995, 11423, 12076, 12974, 13438

Offset: 1

Gerasimov Sergey, Apr 15 2012

nonn

Numbers k such that the sum of first n nonnegative noncomposite numbers is equal to 3k.

			1/3 + 2/3 = 1, 1/3 + 2/3 + 3/3 = 2, 1/3 + 2/3 + 3/3 + 5/3 + 7/3 = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007504, A154293, A158611.

s = 1; t = {0}; Do[s = s + Prime[n]; If[Mod[s, 3] == 0, AppendTo[t, s/3]], {n, 200}]; t (* T. D. Noe, Apr 18 2012 *)
Select[Accumulate[Join[{0,1/3},Prime[Range[200]]/3]],IntegerQ] (* Harvey P. Dale, Mar 06 2016 *)

Definition corrected by Harvey P. Dale, Mar 06 2016

A193470 Square array A(n,k) (n>=1, k>=0) read by antidiagonals: A(n,0) = 0 and A(n,k) is the least integer > A(n,k-1) that can be expressed as a triangular number divided by n.

Original entry on oeis.org

0, 0, 1, 0, 3, 3, 0, 1, 5, 6, 0, 7, 2, 14, 10, 0, 2, 9, 5, 18, 15, 0, 1, 3, 30, 7, 33, 21, 0, 3, 6, 9, 34, 12, 39, 28, 0, 15, 4, 11, 11, 69, 15, 60, 36, 0, 4, 17, 13, 13, 21, 75, 22, 68, 45, 0, 1, 5, 62, 15, 20, 24, 124, 26, 95, 55, 0, 5, 12, 17, 66, 30, 35, 38, 132, 35, 105, 66

Offset: 1

Peter Luschny, Jul 27 2011

			n\k  0   1   2    3    4     5     6     7
------------------------------------------
1 |  0   1   3    6   10    15    21    28    A000217
2 |  0   3   5   14   18    33    39    60    A074378
3 |  0   1   2    5    7    12    15    22    A001318
4 |  0   7   9   30   34    69    75   124    A154260
5 |  0   2   3    9   11    21    24    38    A057569
6 |  0   1   6   11   13    20    35    46    A154293
7 |  0   3   4   13   15    30    33    54    A057570
8 |  0  15  17   62   66   141   147   252    A157716

Cf. A061782, A011772.

A193470_rect := proc(n,k) local j,i,L; L := NULL; j := 0; while nops([L]) < k do add(i/n, i=1..j); if type(%,integer) then L := L,% fi; j := j+1 od; L end:
seq(print(A193470_rect(n, 12)),n = 1..8);

a[, 0] = 0; a[n, k_] := a[n, k] = For[j = a[n, k-1]+1, True, j++, If[Reduce[m > 0 && j == m(m+1)/(2n), m, Integers] =!= False, Return[j]]]; Table[a[n-k, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 07 2016 *)

A287733 First differences of A069497.

Original entry on oeis.org

6, 30, 30, 12, 42, 90, 66, 24, 78, 150, 102, 36, 114, 210, 138, 48, 150, 270, 174, 60, 186, 330, 210, 72, 222, 390, 246, 84, 258, 450, 282, 96, 294, 510, 318, 108, 330, 570, 354, 120, 366, 630, 390, 132, 402, 690, 426, 144, 438, 750, 462, 156, 474, 810, 498, 168, 510, 870, 534

Offset: 1

Greg Huber, May 30 2017

First differences of the subsequence of triangular numbers that are divisible by 6.

By definition, these numbers are themselves divisible by 6.

			The first triangular number divisible by 6 is 6, and the second triangular number divisible by 6 is 36.  Therefore a(2) = 36 - 6 = 30. (The zeroth triangular number divisible by 6 is taken to be 0.)

Cf. A069497, A108782, A154293.

S:= [seq(seq((12*i+j)*(12*i+j+1)/2, j=[0,3,8,11]), i=0..50)]:
S[2..-1]-S[1..-2]; # Robert Israel, May 30 2017

Differences@ Select[Array[# (# + 1)/2 &, 180, 0], Mod[#, 6] == 0 &] (* Robert G. Wilson v, May 30 2017 *)
Differences[Select[Accumulate[Range[0, 209]], Divisible[#, 6] &]] (* Alonso del Arte, May 31 2017 *)

G.f.: 6*(x^2+4*x+1)*(x^2-x+1)/((x-1)^2*(x^2+1)^2). - Robert Israel, May 30 2017

More terms from Robert G. Wilson v, May 30 2017

A164619 Integers of the form A164577(k)/3.

Original entry on oeis.org

4, 15, 54, 75, 132, 169, 320, 459, 735, 847, 1104, 1250, 1764, 2175, 2904, 3179, 3780, 4107, 5200, 6027, 7425, 7935, 9024, 9604, 11492, 12879, 15162, 15979, 17700, 18605, 21504, 23595, 26979, 28175, 30672, 31974, 36100, 39039, 43740, 45387, 48804

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

The sequence members are the third of the average of a set of smallest cubes, if integer.

			A third of the average of the first cube, A164577(1)/3=1/3, is not an integer and does not contribute to the sequence.
A third of the average of the first two cubes, A164577(2)/3=4, is an integer and defines a(1)=4 of the sequence.

Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1,2,-4,2,2,-4,2,-1,2,-1,-1,2,-1).

Cf. A050248, A051456, A078617, A078618, A136116, A154293, A164286, A164576, A164577, A164578, A164579.

s=0;lst={};Do[a=(s+=(n^3)/3)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n,2*5!}]; lst
LinearRecurrence[{2,-1,-1,2,-1,2,-4,2,2,-4,2,-1,2,-1,-1,2,-1},{4,15,54,75,132,169,320,459,735,847,1104,1250,1764,2175,2904,3179,3780},50] (* Harvey P. Dale, Apr 06 2016 *)

Vec(x*(x^14 +x^13 +16*x^12 +10*x^11 +47*x^10 -22*x^9 +61*x^8 +10*x^7 +88*x^6 +8*x^5 +43*x^4 -14*x^3 +28*x^2 +7*x +4) / ((x -1)^4*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^2) + O(x^100)) \\ Colin Barker, Oct 27 2014

a(n) = +2*a(n-1) -a(n-2) -a(n-3) +2*a(n-4) -a(n-5) +2*a(n-6) -4*a(n-7) +2*a(n-8) +2*a(n-9) -4*a(n-10) +2*a(n-11) -a(n-12) +2*a(n-13) -a(n-14) -a(n-15) +2*a(n-16) -a(n-17). - R. J. Mathar, Jan 25 2011

G.f.: x*(x^14 +x^13 +16*x^12 +10*x^11 +47*x^10 -22*x^9 +61*x^8 +10*x^7 +88*x^6 +8*x^5 +43*x^4 -14*x^3 +28*x^2 +7*x +4) / ((x -1)^4*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^2). - Colin Barker, Oct 27 2014

Edited by R. J. Mathar, Aug 20 2009

cp's OEIS Frontend

A182155 Integers of the form: 0/3 + 1/3 + 2/3 + 3/3 + 5/3 + 7/3 + 11/3 + 13/3 + 17/3 + ....

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A193470 Square array A(n,k) (n>=1, k>=0) read by antidiagonals: A(n,0) = 0 and A(n,k) is the least integer > A(n,k-1) that can be expressed as a triangular number divided by n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A287733 First differences of A069497.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A164619 Integers of the form A164577(k)/3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions